- Email: [email protected]
Anderson impurity model: Double occupancy contribution to the magnetic susceptibility
ARTICLE IN PRESS Physica B 404 (2009) 2865–2867
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Anderson impurity model: Double occupancy contribution to the magnetic susceptibility S. Jaroszewicz, P. Roura-Bas ´mico Constituyentes, Comisio ´n Nacional de Energı´a Ato ´mica, Buenos Aires, Argentina Dpto. de Fı´sica, Centro Ato
a r t i c l e in fo
PACS: 70.7 39.4.w Keywords: Magnetic susceptibility Anderson model Non-crossing approximation
abstract After solving the single Anderson impurity model (SIAM) within the non-crossing approximation with a 2 finite Coulomb repulsion, U, and vertex corrections (NCAf v), we focus on the magnetic susceptibility. Using the same diagrammatic expansion the susceptibility can be dressed with two factors, namely, the double state occupancy and the vertex corrections. In this work we analyse the effect of double occupancy on the dynamic and static susceptibility as a function of U and on the degeneracy of the total impurity angular moment, S. & 2009 Elsevier B.V. All rights reserved.
1. Introduction The spin–spin correlation functions are directly measurable by inelastic neutron scattering. The differential cross-section of the neutrons at energy o and wave vector q is proportional to the imaginary part of the Fourier image of the dynamical spin–spin correlation function; wI ðT; o; kÞ=o. For a magnetic impurity (k ¼ 0), which couples to a conduction band, the dynamical susceptibility becomes a single function of temperature, T, and energy o, wðT; oÞ. The static susceptibility, [wR ðT; 0Þ ¼ C=T], at low temperatures of strongly correlated magnetic impurities coupled to conduction electrons does not follow Curie’s law, indicating a total or partial quenching of the impurity spin moment, S. Magnetic impurities embedded in a bath of conduction electrons are commonly described by the single Anderson impurity model (SIAM) [1]. The singly occupied states contribution to the susceptibility within the SIAM using the non-crossing approximation (NCA) in the infinite Coulomb repulsion U, was early studied by Bickers [4], and also by Anders in the same infinite-U limit but within the post-NCA theory (NCA1 ) [5,6]. In the finite-U limit and considering vertex corrections, the same single occupied contribution was analysed by Schiller [7], for different U values and N ¼ 2J þ 1 ¼ 6, appropriate for Ce and Yb ions. For the SIAM and within the NCA approximation, the main contribution to the magnetic susceptibility comes from the singly occupied states. The contribution of the doubly occupied states is found to be small, as compared to the contribution of singly
Corresponding author. Tel.: +54 11 67727102; fax: +54 11 67727121.
E-mail address: [email protected] (P. Roura-Bas). 0921-4526/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2009.06.105
occupied ones. It is actually proportional to ðN 1Þ=U, so that its influence only appears for small values of U and large degeneracy. This is not the case for Ce and Yb ions due to the large characteristic value of their Coulomb repulsion, U, which lies around 6 eV. Single Co impurities adsorbed on different noble metal surfaces seem to be a realization of the finite U limit due to its impurity number occupation, which is around 7, to the strong Hund’s coupling which produces an effective spin S ¼ 32 (N ¼ 4) and, to the value of the Coulomb’s repulsion, estimated to lie between 2.5 and 4 eV [8]. It is interesting to analyse this additional contribution in this parameters’ range. In order to study the influence of the doubly occupied states on the susceptibility, we focus in this work on a fourfold degenerate impurity magnetic moment described by the Anderson model 2 within the NCAf v scheme [2,3], for different values of U.
2. Method of calculation The degenerate Anderson impurity model with finite U is described by the Hamiltonian X X ^ ¼ ^m ks N^ ks þ d H N m
ks
þU
X mon
^ mN ^n þV N
X y ðd^ m c^ ks þ H:c:Þ,
(1)
ksm
where the indices m; n label the quantum numbers of the impurity y y levels. The operators c^ ks ; d^ m create a conduction electron and a localized one with energies ks and d , respectively. Operators ^ m ¼ d^ y d^ m are the cks -number and d-number ^ ks ¼ c^ ks cks and N N m operators, respectively.
ARTICLE IN PRESS 2866
S. Jaroszewicz, P. Roura-Bas / Physica B 404 (2009) 2865–2867
The physics of the model is completely determined by the set of parameters d ; N ¼ 2S þ 1 (degeneracy of the total spin moment), the Coulomb on-site repulsion U and the hybridization between conduction and localized electrons given by X 2 GðÞ ¼ p V dð ks Þ. (2) ks
In the auxiliary particle representation, the Hamiltonian becomes X
ks N^ ks þ d
ks
þV
X
y s^ m s^ m þ ð2d þ UÞ
m
X
y a^ mn a^ mn
mon
X y X y ðs^ m b^ c^ ks þ H:c:Þ þ V ða^ mn s^ m c^ ks þ H:c:Þ, ksm
(3)
ks;mon
^ s^ and a^ represent the vacuum, single where the operators b, and double occupied states, respectively. We solve the Hamiltonian diagrammatically within the non-crossing approximation 2 including vertex corrections as implemented in the NCAf v method [2,3]. In order to obtain the magnetic properties of the model, we solve the spin–spin correlation function using the same noncrossing diagrammatic procedure. The main contribution comes from the single occupied states s^ m , but there are two contributions that can be added, namely, the vertex correction and the contribution that comes from the double occupied states a^ mn . The non-crossing diagrams, without vertex corrections, for the dynamic susceptibility are presented in Fig. 1 and the corresponding analytic expression for the imaginary part is given by
wI ðT; oÞ ¼ p
Z
þ1
dn
rm ðnÞ
ðrm ðn oÞ rm ðn þ oÞÞ ebn Z þ1 r 0 ðnÞ þ ðN 1Þ dn mmbn ðrmm0 ðn oÞ rmm0 ðn þ oÞÞ, e 1 (4) 1
where rm ðoÞ, m ¼ m; mm0 , are the auxiliary particle spectral functions defined by rm ðoÞ ¼ 1=pGIm ðo þ i0Þ. The exponential divergencies that appear due to the statistical Boltzmann factors are compensated by the threshold behaviour of the corresponding auxiliary particle spectral functions rm ðoÞ in the integrands [12,13]. The real part of the dynamic susceptibility follows directly from the imaginary one through the Kramers–Kronig relations and the static susceptibility, wðTÞ, is obtained by setting o ¼ 0 in the resulting expression, 1
wR ðT; oÞ ¼ P p
Z
In this section, we present the numerical results for the dynamic and static magnetic susceptibilities obtained from the above mentioned expressions within the SIAM. As stated above, we consider the fourfold degenerate (N ¼ 4) case. We take a rectangular model for the hybridization function GðoÞ ¼ G0 yðD jojÞ with a conduction band of half width D. The parameters for the numerical calculations are within the Kondo regime, that is d bG0 . In the Kondo regime, the Anderson’s model susceptibility maps smoothly onto the Kondo model susceptibility with an effective Kondo temperature, T K . We set d ¼ 2 eV and G0 ¼ 0:3 eV, with D ¼ 5 eV. The Coulomb repulsion U is varied from 2.5 to 5 eV. Fig. 2 shows the high energy range for the imaginary part of the dynamic susceptibility in the case U ¼ 2:5 eV. Within this range of parameters the Kondo temperature of the system is T K ¼ 94 K [9,4]. The calculation is performed at T ¼ 0:1T K . Fig. 3 shows the real part of the dynamic susceptibility for the same set of parameters as in Fig. 2. As expected, in both, real and imaginary spectra of the dynamic susceptibility, the double occupied state contribution shows up in the high energy range. The double occupied states appear as symmetric peaks above joj=DU=2D. These additional peaks are not related to the inelastic ones, that appear at low temperature and for large degeneracies (N ¼ 6) [4,10,5,11]. While the dynamic susceptibility is sensible to the additional contribution, the static one does not exhibit significant changes. This follows from the way in which the static susceptibility is obtained, namely, as the limit at low energies, (o ¼ 0), of the real part of the dynamic one, as it has been mentioned above. In Fig. 4 we show the static susceptibility as a function of T=T K for the same set of parameters as in Fig. 2. The susceptibility presents the usual behaviour, that is, saturation at temperatures less than T K . Quantitatively, we obtain an increase in the static susceptibility of around 1% due to the additional contribution. In order to analyse the evolution of the additional contribution for increasing values of the Coulomb repulsion, we show in Fig. 5 the imaginary part of the dynamic susceptibility in units of g mB for three values of U, namely, U ¼ 2:5, 3.5 and 5:0 eV. The
0.1 0.08 0.06 0.04
wI ðT; o0 Þ do0 , 0 1 o o 1
(5)
wðTÞ ¼ wR ðT; 0Þ.
(6)
χI (ω)/μeff (j)
^ ¼ H
3. Numerical results and discussion
0.02 0 -0.02 -0.04
ipn m
ipn
-0.06
mm’
-0.08 -0.1
mm’ m ipn + iwn
ipn + iwn
Fig. 1. NCA diagrams for the dynamical magnetic susceptibility. The full and curly lines represent the single and double occupied state propagators. Diagram (a) and (b) show the single and double occupied contribution to susceptibility.
−1
−0.8 −0.6 −0.4 −0.2
0
0.2
0.4
0.6
0.8
1
ω/D Fig. 2. High energy range for the imaginary part of the dynamic susceptibility in units of g mB . Dashed line is used for the single occupied contribution, while the solid line shows the result including the contribution of the doubly occupied states. Parameters: N ¼ 4, ed ¼ 2, U ¼ 2:5 and G ¼ 0:3. Energies are in eV.
ARTICLE IN PRESS S. Jaroszewicz, P. Roura-Bas / Physica B 404 (2009) 2865–2867
2867
0.3
0.02
U = 2.5 U = 3.5 U = 5.0
0.2
0
χI (ω)/μeff (j)
χR (ω)/μeff (j)
0.1
-0.02
-0.04
0
-0.1
-0.06 -0.2
-0.08 -1
-0.8
-0.6
-0.4
-0.2
0 ω/D
0.2
0.4
0.6
0.8
1
Fig. 3. High energy range for the real part of the dynamic susceptibility. Parameters are as Fig. 2.
-2
-1.6
-1.2
-0.4
0 ω/D
0.4
0.8
1.2
1.6
2
Fig. 5. High energy range of the imaginary part of the dynamic susceptibility in units of g mB for different values of the Coulomb repulsion, U, and at T ¼ 0:1T K . Parameters are as Fig. 2. Energies are in eV.
12
2
Anderson model, within the NCAf v approximation in the Kondo regime for the particular case of a fourfold degeneracy and for different values of the Coulomb repulsion. This particular set of parameters are reasonable ones to study the behaviour of Co impurities adsorbed on different noble metal surfaces. The additional contribution is relevant in the high energy spectra of the dynamical susceptibility, while the effect on the static one is small.
10 TK = 94 K 8
χ (Τ)
-0.8
6
Acknowledgements 4
The authors are very grateful to Dr. A. Llois for many fruitful discussions on the subject of this work and careful reading of the manuscript. This work was partially funded by UBACyT-X115, PICT-033304 and PIP-CONICET 6016.
2
References
0 0.1
1
10
100
T/TK Fig. 4. Temperature dependence of the static magnetic susceptibility in units of 2 g mB computed within the NCAf v with N ¼ 4. Parameters are as Fig. 2.
temperature is T ¼ 0:1T K in all cases. It is clear that the double state contribution decreases very fast with growing U values. The peaks due to the finite value of U move to higher energies as U increases.
4. Summary and discussion In summary, we have studied the contribution of double occupied states on the dynamic and static susceptibilities of the
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
P.W. Anderson, Phys. Rev. 124 (1961) 41. J. Otsuki, Y. Kuramoto, J. Phys. Soc. Japan 75 (2006) 064707. O. Sakai, Y. Shimizu, Y. Kaneta, J. Phys. Soc. Japan 74 (2005) 2517. N.E. Bickers, Rev. Modern Phys. 59 (1987) 845; N.E. Bickers, D.L. Cox, J.W. Wilkins, Phys. Rev. B 36 (1987) 2036. F.B. Anders, J. Phys. Condens. Matter 7 (1995) 2801. N. Grewe, S. Schmitt, T. Jabben, F.B. Anders, J. Phys. Condens. Matter 20 (2008) 365217. A. Schiller, V. Zevin, Phys. Rev. B 47 (1993) 9297. A.B. Shick, A.I. Lichtenstein, J. Phys.: Condens. Matter 20 (2008) 015002. We use the definition of the Kondo temperature as given in Ref. [4], it is as the energy position of the Kondo resonance in the spectral function. Y. Kuramoto, H. Kojima, Z. Phys. B Condens. Matter 57 (1984) 95. A. Hewson, The Kondo Problems to Heavy Fermions, Cambridge University Press, Cambridge, 1993, p. 212. T.A. Costi, J. Kroha, P. Wo¨lfle, Phys. Rev. B 53 (1996) 1850. For the numerical treatment, these divergencies can be explicitly absorbed by ˜ m ðoÞ formulating the self-consistency of the NCA in terms of the functions r ˜ m ðoÞ, where f ðoÞ is the Fermi function. which are defined via rm ðoÞ ¼ f ðoÞr
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Anderson impurity model: Double occupancy contribution to the magnetic susceptibility S. Jaroszewicz, P. Roura-Bas ´mico Constituyentes, Comisio ´n Nacional de Energı´a Ato ´mica, Buenos Aires, Argentina Dpto. de Fı´sica, Centro Ato
a r t i c l e in fo
PACS: 70.7 39.4.w Keywords: Magnetic susceptibility Anderson model Non-crossing approximation
abstract After solving the single Anderson impurity model (SIAM) within the non-crossing approximation with a 2 finite Coulomb repulsion, U, and vertex corrections (NCAf v), we focus on the magnetic susceptibility. Using the same diagrammatic expansion the susceptibility can be dressed with two factors, namely, the double state occupancy and the vertex corrections. In this work we analyse the effect of double occupancy on the dynamic and static susceptibility as a function of U and on the degeneracy of the total impurity angular moment, S. & 2009 Elsevier B.V. All rights reserved.
1. Introduction The spin–spin correlation functions are directly measurable by inelastic neutron scattering. The differential cross-section of the neutrons at energy o and wave vector q is proportional to the imaginary part of the Fourier image of the dynamical spin–spin correlation function; wI ðT; o; kÞ=o. For a magnetic impurity (k ¼ 0), which couples to a conduction band, the dynamical susceptibility becomes a single function of temperature, T, and energy o, wðT; oÞ. The static susceptibility, [wR ðT; 0Þ ¼ C=T], at low temperatures of strongly correlated magnetic impurities coupled to conduction electrons does not follow Curie’s law, indicating a total or partial quenching of the impurity spin moment, S. Magnetic impurities embedded in a bath of conduction electrons are commonly described by the single Anderson impurity model (SIAM) [1]. The singly occupied states contribution to the susceptibility within the SIAM using the non-crossing approximation (NCA) in the infinite Coulomb repulsion U, was early studied by Bickers [4], and also by Anders in the same infinite-U limit but within the post-NCA theory (NCA1 ) [5,6]. In the finite-U limit and considering vertex corrections, the same single occupied contribution was analysed by Schiller [7], for different U values and N ¼ 2J þ 1 ¼ 6, appropriate for Ce and Yb ions. For the SIAM and within the NCA approximation, the main contribution to the magnetic susceptibility comes from the singly occupied states. The contribution of the doubly occupied states is found to be small, as compared to the contribution of singly
Corresponding author. Tel.: +54 11 67727102; fax: +54 11 67727121.
E-mail address: [email protected] (P. Roura-Bas). 0921-4526/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2009.06.105
occupied ones. It is actually proportional to ðN 1Þ=U, so that its influence only appears for small values of U and large degeneracy. This is not the case for Ce and Yb ions due to the large characteristic value of their Coulomb repulsion, U, which lies around 6 eV. Single Co impurities adsorbed on different noble metal surfaces seem to be a realization of the finite U limit due to its impurity number occupation, which is around 7, to the strong Hund’s coupling which produces an effective spin S ¼ 32 (N ¼ 4) and, to the value of the Coulomb’s repulsion, estimated to lie between 2.5 and 4 eV [8]. It is interesting to analyse this additional contribution in this parameters’ range. In order to study the influence of the doubly occupied states on the susceptibility, we focus in this work on a fourfold degenerate impurity magnetic moment described by the Anderson model 2 within the NCAf v scheme [2,3], for different values of U.
2. Method of calculation The degenerate Anderson impurity model with finite U is described by the Hamiltonian X X ^ ¼ ^m ks N^ ks þ d H N m
ks
þU
X mon
^ mN ^n þV N
X y ðd^ m c^ ks þ H:c:Þ,
(1)
ksm
where the indices m; n label the quantum numbers of the impurity y y levels. The operators c^ ks ; d^ m create a conduction electron and a localized one with energies ks and d , respectively. Operators ^ m ¼ d^ y d^ m are the cks -number and d-number ^ ks ¼ c^ ks cks and N N m operators, respectively.
ARTICLE IN PRESS 2866
S. Jaroszewicz, P. Roura-Bas / Physica B 404 (2009) 2865–2867
The physics of the model is completely determined by the set of parameters d ; N ¼ 2S þ 1 (degeneracy of the total spin moment), the Coulomb on-site repulsion U and the hybridization between conduction and localized electrons given by X 2 GðÞ ¼ p V dð ks Þ. (2) ks
In the auxiliary particle representation, the Hamiltonian becomes X
ks N^ ks þ d
ks
þV
X
y s^ m s^ m þ ð2d þ UÞ
m
X
y a^ mn a^ mn
mon
X y X y ðs^ m b^ c^ ks þ H:c:Þ þ V ða^ mn s^ m c^ ks þ H:c:Þ, ksm
(3)
ks;mon
^ s^ and a^ represent the vacuum, single where the operators b, and double occupied states, respectively. We solve the Hamiltonian diagrammatically within the non-crossing approximation 2 including vertex corrections as implemented in the NCAf v method [2,3]. In order to obtain the magnetic properties of the model, we solve the spin–spin correlation function using the same noncrossing diagrammatic procedure. The main contribution comes from the single occupied states s^ m , but there are two contributions that can be added, namely, the vertex correction and the contribution that comes from the double occupied states a^ mn . The non-crossing diagrams, without vertex corrections, for the dynamic susceptibility are presented in Fig. 1 and the corresponding analytic expression for the imaginary part is given by
wI ðT; oÞ ¼ p
Z
þ1
dn
rm ðnÞ
ðrm ðn oÞ rm ðn þ oÞÞ ebn Z þ1 r 0 ðnÞ þ ðN 1Þ dn mmbn ðrmm0 ðn oÞ rmm0 ðn þ oÞÞ, e 1 (4) 1
where rm ðoÞ, m ¼ m; mm0 , are the auxiliary particle spectral functions defined by rm ðoÞ ¼ 1=pGIm ðo þ i0Þ. The exponential divergencies that appear due to the statistical Boltzmann factors are compensated by the threshold behaviour of the corresponding auxiliary particle spectral functions rm ðoÞ in the integrands [12,13]. The real part of the dynamic susceptibility follows directly from the imaginary one through the Kramers–Kronig relations and the static susceptibility, wðTÞ, is obtained by setting o ¼ 0 in the resulting expression, 1
wR ðT; oÞ ¼ P p
Z
In this section, we present the numerical results for the dynamic and static magnetic susceptibilities obtained from the above mentioned expressions within the SIAM. As stated above, we consider the fourfold degenerate (N ¼ 4) case. We take a rectangular model for the hybridization function GðoÞ ¼ G0 yðD jojÞ with a conduction band of half width D. The parameters for the numerical calculations are within the Kondo regime, that is d bG0 . In the Kondo regime, the Anderson’s model susceptibility maps smoothly onto the Kondo model susceptibility with an effective Kondo temperature, T K . We set d ¼ 2 eV and G0 ¼ 0:3 eV, with D ¼ 5 eV. The Coulomb repulsion U is varied from 2.5 to 5 eV. Fig. 2 shows the high energy range for the imaginary part of the dynamic susceptibility in the case U ¼ 2:5 eV. Within this range of parameters the Kondo temperature of the system is T K ¼ 94 K [9,4]. The calculation is performed at T ¼ 0:1T K . Fig. 3 shows the real part of the dynamic susceptibility for the same set of parameters as in Fig. 2. As expected, in both, real and imaginary spectra of the dynamic susceptibility, the double occupied state contribution shows up in the high energy range. The double occupied states appear as symmetric peaks above joj=DU=2D. These additional peaks are not related to the inelastic ones, that appear at low temperature and for large degeneracies (N ¼ 6) [4,10,5,11]. While the dynamic susceptibility is sensible to the additional contribution, the static one does not exhibit significant changes. This follows from the way in which the static susceptibility is obtained, namely, as the limit at low energies, (o ¼ 0), of the real part of the dynamic one, as it has been mentioned above. In Fig. 4 we show the static susceptibility as a function of T=T K for the same set of parameters as in Fig. 2. The susceptibility presents the usual behaviour, that is, saturation at temperatures less than T K . Quantitatively, we obtain an increase in the static susceptibility of around 1% due to the additional contribution. In order to analyse the evolution of the additional contribution for increasing values of the Coulomb repulsion, we show in Fig. 5 the imaginary part of the dynamic susceptibility in units of g mB for three values of U, namely, U ¼ 2:5, 3.5 and 5:0 eV. The
0.1 0.08 0.06 0.04
wI ðT; o0 Þ do0 , 0 1 o o 1
(5)
wðTÞ ¼ wR ðT; 0Þ.
(6)
χI (ω)/μeff (j)
^ ¼ H
3. Numerical results and discussion
0.02 0 -0.02 -0.04
ipn m
ipn
-0.06
mm’
-0.08 -0.1
mm’ m ipn + iwn
ipn + iwn
Fig. 1. NCA diagrams for the dynamical magnetic susceptibility. The full and curly lines represent the single and double occupied state propagators. Diagram (a) and (b) show the single and double occupied contribution to susceptibility.
−1
−0.8 −0.6 −0.4 −0.2
0
0.2
0.4
0.6
0.8
1
ω/D Fig. 2. High energy range for the imaginary part of the dynamic susceptibility in units of g mB . Dashed line is used for the single occupied contribution, while the solid line shows the result including the contribution of the doubly occupied states. Parameters: N ¼ 4, ed ¼ 2, U ¼ 2:5 and G ¼ 0:3. Energies are in eV.
ARTICLE IN PRESS S. Jaroszewicz, P. Roura-Bas / Physica B 404 (2009) 2865–2867
2867
0.3
0.02
U = 2.5 U = 3.5 U = 5.0
0.2
0
χI (ω)/μeff (j)
χR (ω)/μeff (j)
0.1
-0.02
-0.04
0
-0.1
-0.06 -0.2
-0.08 -1
-0.8
-0.6
-0.4
-0.2
0 ω/D
0.2
0.4
0.6
0.8
1
Fig. 3. High energy range for the real part of the dynamic susceptibility. Parameters are as Fig. 2.
-2
-1.6
-1.2
-0.4
0 ω/D
0.4
0.8
1.2
1.6
2
Fig. 5. High energy range of the imaginary part of the dynamic susceptibility in units of g mB for different values of the Coulomb repulsion, U, and at T ¼ 0:1T K . Parameters are as Fig. 2. Energies are in eV.
12
2
Anderson model, within the NCAf v approximation in the Kondo regime for the particular case of a fourfold degeneracy and for different values of the Coulomb repulsion. This particular set of parameters are reasonable ones to study the behaviour of Co impurities adsorbed on different noble metal surfaces. The additional contribution is relevant in the high energy spectra of the dynamical susceptibility, while the effect on the static one is small.
10 TK = 94 K 8
χ (Τ)
-0.8
6
Acknowledgements 4
The authors are very grateful to Dr. A. Llois for many fruitful discussions on the subject of this work and careful reading of the manuscript. This work was partially funded by UBACyT-X115, PICT-033304 and PIP-CONICET 6016.
2
References
0 0.1
1
10
100
T/TK Fig. 4. Temperature dependence of the static magnetic susceptibility in units of 2 g mB computed within the NCAf v with N ¼ 4. Parameters are as Fig. 2.
temperature is T ¼ 0:1T K in all cases. It is clear that the double state contribution decreases very fast with growing U values. The peaks due to the finite value of U move to higher energies as U increases.
4. Summary and discussion In summary, we have studied the contribution of double occupied states on the dynamic and static susceptibilities of the
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
P.W. Anderson, Phys. Rev. 124 (1961) 41. J. Otsuki, Y. Kuramoto, J. Phys. Soc. Japan 75 (2006) 064707. O. Sakai, Y. Shimizu, Y. Kaneta, J. Phys. Soc. Japan 74 (2005) 2517. N.E. Bickers, Rev. Modern Phys. 59 (1987) 845; N.E. Bickers, D.L. Cox, J.W. Wilkins, Phys. Rev. B 36 (1987) 2036. F.B. Anders, J. Phys. Condens. Matter 7 (1995) 2801. N. Grewe, S. Schmitt, T. Jabben, F.B. Anders, J. Phys. Condens. Matter 20 (2008) 365217. A. Schiller, V. Zevin, Phys. Rev. B 47 (1993) 9297. A.B. Shick, A.I. Lichtenstein, J. Phys.: Condens. Matter 20 (2008) 015002. We use the definition of the Kondo temperature as given in Ref. [4], it is as the energy position of the Kondo resonance in the spectral function. Y. Kuramoto, H. Kojima, Z. Phys. B Condens. Matter 57 (1984) 95. A. Hewson, The Kondo Problems to Heavy Fermions, Cambridge University Press, Cambridge, 1993, p. 212. T.A. Costi, J. Kroha, P. Wo¨lfle, Phys. Rev. B 53 (1996) 1850. For the numerical treatment, these divergencies can be explicitly absorbed by ˜ m ðoÞ formulating the self-consistency of the NCA in terms of the functions r ˜ m ðoÞ, where f ðoÞ is the Fermi function. which are defined via rm ðoÞ ¼ f ðoÞr